About the combination of tossing three dices

[KFC]

Tossing three dices, how many combination of the sum (from 3 to 18)? I just found a general expression to find out such combination, which is

(x + x^2 + x^3 + x^4 + x^5 + x^6)^3

the total combination of given sum for three dices is the coefficient of the corresponding term,

for example, for the sum 10, the total combination is 27, which is the coefficient of the 10th term (x^10).

I wonder how people find such expression? Why the combination is just the coefficient? Someone told me it is just a coincident, which is hard to convince me.

 

[Office_Shredder]

This is a simple example of what's generally called a generating function.

Let's look at a simpler example. You flip two coins, for each flip you add 1 if you get heads, or 2 if you get tails. Now let's consider multiplication:

(x+x²)(x+x²).

Each term x+x² is going to fill in for one coin flip. The power of x is going to be the value of the flip, 1 or 2. To count the number of ways that flips can add up to a 3, I need to pick a value of each coin flip independently so that everything adds up to 3. This is the same as picking a power of x in each parentheses so that the powers of x add up to 3, and then multiplying all the x's together. But when you expand, the way multiplication works to find the coefficient of x³ is exactly the same as counting how many ways there are to pick a power of x from each polynomial so that the powers add up to 3.

It's the same principle for the dice. To see how many ways there are for the dice to add up to 4, you have to pick a value for each die, which is the same as picking a power of x from each polynomial (x+x²...+x⁶). And when you multiply, the coefficient of x⁴ is exactly the number of ways to do this

 

[KFC]

Excellent explanation. Thanks a lot.

This is a simple example of what's generally called a generating function.

Let's look at a simpler example. You flip two coins, for each flip you add 1 if you get heads, or 2 if you get tails. Now let's consider multiplication:

(x+x²)(x+x²).

Each term x+x² is going to fill in for one coin flip. The power of x is going to be the value of the flip, 1 or 2. To count the number of ways that flips can add up to a 3, I need to pick a value of each coin flip independently so that everything adds up to 3. This is the same as picking a power of x in each parentheses so that the powers of x add up to 3, and then multiplying all the x's together. But when you expand, the way multiplication works to find the coefficient of x³ is exactly the same as counting how many ways there are to pick a power of x from each polynomial so that the powers add up to 3.

It's the same principle for the dice. To see how many ways there are for the dice to add up to 4, you have to pick a value for each die, which is the same as picking a power of x from each polynomial (x+x²...+x⁶). And when you multiply, the coefficient of x⁴ is exactly the number of ways to do this

 

[Mark44]

Tossing three dices ...

Very minor point - Dice is the plural of die, so one die, two dice. One mouse, two mice. One louse, two lice. One house, two houses. English is nothing if not inconsistent.

 

[HallsofIvy]

In other words, English is consistently inconsistent!

 

[CRGreathouse]

In other words, English is consistently inconsistent!

English has something of a fractal grammar. If you learn the rule, "plurals end with s" then you can form the correct plural for the majority of words and an understandable 'plural' for all. If you pick up the rule on -es as well you'll do much better. There are a number of semiregular endings as well: -us to -i, -um to -a, and -x to -en, for example; you'll make mistakes if you assume that this is always the case (octopus -> octopuses, not *octopi), but you'll improve on the whole. (Of course there's a recent trend to pluralize words with -us as -i even when this has not historically been acceptable in English... that's a different issue.) If you then learn the irregular or ablaut forms of a dozen common words you'll be that much better... etc.